Edgar Chávez

Edgar Chávez, Ana C. Chávez-Cáliz, Jorge L. López-López

Given a collection $\{Z_1,Z_2,\ldots,Z_m\}$ of $n$-sided polygons in the plane and a query polygon $W$ we give algorithms to find all $Z_\ell$ such that $W=f(Z_\ell)$ with $f$ an unknown similarity transformation in time independent of the size of the collection. If $f$ is a known affine transformation, we show how to find all $Z_\ell$ such that $W=f(Z_\ell)$ in $O(n+\log(m))$ time. For a pair $W,W^\prime$ of polygons we can find all the pairs $Z_\ell,Z_{\ell^\prime}$ such that $W=f(Z_\ell)$ and $W^\prime=f(Z_{\ell^\prime})$ for an unknown affine transformation $f$ in $O(m+n)$ time. For the case of triangles we also give bounds for the problem of matching triangles with variable vertices, which is equivalent to affine matching triangles in noisy conditions.

Rafael G. Campos, J. Rico-Melgoza, Edgar Chávez

In recent years there has been a growing interest in the fractional Fourier transform driven by its large number of applications. The literature in this field follows two main routes. On the one hand, the areas where the ordinary Fourier transform has been applied are being revisited to use this intermediate time-frequency representation of signals, and on the other hand, fast algorithms for numerical computation of the fractional Fourier transform are devised. In this paper we derive a Gaussian-like quadrature of the continuous fractional Fourier transform. This quadrature is given in terms of the Hermite polynomials and their zeros. By using some asymptotic formulas, we rewrite the quadrature as a chirp-fft-chirp transformation, yielding a fast discretization of the fractional Fourier transform and its inverse in closed form. We extend the range of the fractional Fourier transform by considering arbitrary complex values inside the unitary circle and not only at the boundary. We find that, the chirp-fft-chirp transformation evaluated at z=i, becomes a more accurate version of the fft which can be used for non-periodic functions.